Written by:   Randy Giese
Written at:   www.RandyGrams.com
Written on:   September 19, 2011
Language:     VB.NET / Visual Basic 2010
Problem:   Find an eight digit number, using the digits 1-8, in which no digit is repeated, and when that number is multiplied by 9, gives a nine-digit number in which no digit is repeated.
I found this problem in a book titled "ARITHMETRICKS" by JEROME S. MEYER.   The book was published in 1965 by "SCHOLASTIC BOOK SERVICES, New York".   It was published prior to the introduction of the ISBN format.
According to the author, there are 4 numbers that satisfy the criteria explained above in "Problem:".   He was sure there were not any more, so I took that as a challenge and came up with this program.
Here is an example:     57638214 * 9 = 518743926
The first number is 8 digits long and contains all the digits 1-8.   When you multiply it by 9, you get a 9 digit number that contains all of the digits 1-9.   I found 144 such numbers.   This program finds and lists all of those numbers.
Updated on:   September 17, 2012
I reread the problem in the book and found out that I had skipped a step.   It is true that the 144 numbers do what it says above.   However, there was another line farther down that added another condition.   The author stated that when multiplied by 18, the 4 numbers that the "Brilliant and Unknown Mathematician" found also created a 10 digit number with no repeating numbers.
Whoops!   That makes a difference, so I went back to the drawing board.   I took the list of 144 numbers and multiplied them by 18.   That trimmed the list down to 22 numbers but only half of them make the 10 digit numbers as explained in the paragraph above.   The first half of the 22 still make a 9 digit number, that contains all of the digits 1-9.   So to be honest, I only found 11 numbers that meet ALL the criteria.
I added 2 Check Boxes at the top of the form.   The one labeled "8x9 List" will still produce the 144 numbers.   The one labeled "8x18 List" will create the shortened list of 22 numbers.